As a next-to-minimal possibility along these lines, recent years have seen a growing
interest in the topic of three Higgs-doublet models (3HDMs). These models add two extra
scalar doublets
on top of the SM one, thereby conforming to the aesthetic appeal
of having three scalar generations in harmony with the three generations of fermions.
An early proposal by Weinberg [1]} included two discrete symmetries
which yielded flavour conservation. A recent study [2]}, following
earlier work in [3]}, [4]}, [5]}, used a generalized
CP symmetry (called CP4) to constrain the vast parameter space of 3HDM, and was found to
have large regions of parameter space which conformed with experimental constraints in
the flavour and scalar sectors. That model, however, used the liberty in parameters
present in the Yukawa sector to fit all observables, but the FCNC couplings were not necessarily small. Furthermore, it was shown in [6]} that the model contains several regions where top decays to light charged Higgs bosons is in conflict with LHC data. In [7]}, [8]} is was also studied how baryogenesis can be realized via the decay of new TeV-scale Higgs bosons in a 3HDM. Last but not least, in [9]}, a maximally-symmetric version of the 3HDM based on the \(\mathrm {Sp}(6)\)  symmetry was proposed.
